A student wonders how x^(a/b) changes for different multiples of a and b, and what difference it makes if you express that function as the b-th root of x^a. Doctor Ali corroborates the student's suspicion that domains matter -- as does the order of operations.

I'm imagining a function that doubles after a certain interval, then
doubles again in half that time, doubles again in a quarter of the
original interval, etc. Are there functions that behave this way?

Given one function and the result of its composition with a second function, a student
wonders how to recover the second function. Doctor Peterson shows how substitution
solves such problems; Doctor Rick follows up with a method that relies on the inverse
function.

A student seeks clarity on the meaning of f^2(x) and other expressions that juxtapose
functions and exponents. Doctor Vogler empathizes, outlining the differing
interpretations, before recommending contextual clues — and pointing up a
larger lesson about ambiguity.